(* Properties concerning parallel unfold on local environments **************)
-lemma ltpr_ltps_conf: ∀L1,K1,d,e. L1 [d, e] ▶ K1 → ∀L2. L1 ➡ L2 →
- ∃∃K2. L2 [d, e] ▶* K2 & K1 ➡ K2.
+lemma ltpr_ltpss_conf: ∀L1,K1,d,e. L1 ▶* [d, e] K1 → ∀L2. L1 ➡ L2 →
+ ∃∃K2. L2 ▶* [d, e] K2 & K1 ➡ K2.
#L1 #K1 #d #e #H elim H -L1 -K1 -d -e
[ /2 width=3/
| #L1 #I #V1 #X #H
| #L1 #K1 #I #V1 #W1 #e #_ #HVW1 #IHLK1 #X #H
elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12
- elim (tpr_tps_ltpr … HV12 … HVW1 … HK12) -V1 /3 width=5/
+ elim (tpr_tpss_ltpr … HK12 … HV12 … HVW1) -V1 /3 width=5/
| #L1 #K1 #I #V1 #W1 #d #e #_ #HVW1 #IHLK1 #X #H
elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
elim (IHLK1 … HL12) -L1 #K2 #HLK2 #HK12
- elim (tpr_tps_ltpr … HV12 … HVW1 … HK12) -V1 /3 width=5/
+ elim (tpr_tpss_ltpr … HK12 … HV12 … HVW1) -V1 /3 width=5/
]
qed.
-
-lemma ltpr_ltpss_conf: ∀L1,K1,d,e. L1 [d, e] ▶* K1 → ∀L2. L1 ➡ L2 →
- ∃∃K2. L2 [d, e] ▶* K2 & K1 ➡ K2.
-#L1 #K1 #d #e #H @(ltpss_ind … H) -K1 /2 width=3/
-#K #K1 #_ #HK1 #IHK #L2 #HL12
-elim (IHK … HL12) -L1 #K2 #HLK2 #HK2
-elim (ltpr_ltps_conf … HK1 … HK2) -K #K #HK2 #HK1
-lapply (ltpss_trans_eq … HLK2 HK2) -K2 /2 width=3/
-qed.